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icon:: 📖 purchased:: start:: end:: published:: length:: author:: @Bertrand Russell cover:: reading-time:: score:: - Ideas - Backward Journey - The most obvious and easy things in mathematics are not those that come logically at the beginning; they are things that, from the point of view of logical deduction, come somewhere in the middle. - Just as the easiest bodies to see are those that are neither very near nor very far, neither very small nor very great, so the easiest conceptions to grasp are those that are neither very complex nor very simple (using "simple" in a logical sense). - And as we need two sorts of instruments, the telescope and the microscope, for the enlargement of our visual powers, so we need two sorts of instruments for the enlargement of our logical powers, one to take us forward to the higher mathematics, the other to take us backward to the logical foundations of the things that we are inclined to take for granted in mathematics. - We shall find that by analysing our ordinary mathematical notions we acquire fresh insight, new powers, and the means of reaching whole new mathematical subjects by adopting fresh lines of advance after our backward journey. - The other direction, which is less familiar, proceeds, by analysing, to greater and greater abstractness and logical simplicity; instead of asking what can be defined and deduced from what is assumed to begin with, we ask instead what more general ideas and principles can be found, in terms of which what was our starting-point can be defined or deduced. id:: 64e3a7c8-b00e-4df0-a3b4-372c04027308 - It is the fact of pursuing this opposite direction that characterizes mathematical philosophy as opposed to ordinary mathematics. - But it should be understood that the distinction is one, not in the subject matter, but in the state of mind of the investigator. - Highlights -